Please help with this example to apply the laws of logarithms to simplify the function $\displaystyle f(x) =ln \sqrt\frac{9-x^2}{4+x^2}$. Then find it’s derivative.

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- Oct 28th 2006, 07:19 AMYogi_Bear_79laws of logarithms
Please help with this example to apply the laws of logarithms to simplify the function $\displaystyle f(x) =ln \sqrt\frac{9-x^2}{4+x^2}$. Then find it’s derivative.

- Oct 28th 2006, 07:27 AMQuick
I can't help you with the derivative, but here's the simply part:

you have: $\displaystyle \ln\sqrt{\frac{9-x^2}{4+x^2}}$

rewrite: $\displaystyle \ln\left(\left(\frac{9-x^2}{4+x^2}\right)^{\frac{1}{2}}\right)$

Thus: $\displaystyle \frac{1}{2}\ln\frac{9-x^2}{4+x^2}$

rewrite: $\displaystyle \frac{1}{2}\ln\frac{(3+x)(3-x)}{4+x^2}$

It is possible to go farther but I don't know how that might help... - Oct 28th 2006, 07:35 AMSoroban
Hello, Yogi_Bear_79!

Do you*know*the Laws of Logarithms?

Quote:

Apply the laws of logarithms to simplify the function $\displaystyle f(x) = \ln\sqrt\frac{9-x^2}{4+x^2}$

Then find its derivative.

We have: .$\displaystyle f(x) \;= \;\ln\sqrt{\frac{9-x^2}{4+x^2}} \:=\:\ln\!\left(\frac{9-x^2}{4+x^2}\right)^{\frac{1}{2}} \;=\;\frac{1}{2}\cdot\ln\!\left(\frac{9-x^2}{4+x^2}\right)$

Then: .$\displaystyle f(x)\;=\;\frac{1}{2}\bigg[\ln(9-x^2) - \ln(4 + x^2)\bigg] $

Now differentiate it . . .

- Oct 28th 2006, 07:37 AMQuick
- Oct 28th 2006, 11:23 AMtopsquark
- Oct 28th 2006, 12:04 PMearboth